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THEOREM 3.4. Let $(X,T)$ be a topological space and let $A?X$. Then the following conditions are equivalent. (a) The subspace $(A,T_{A})$ is connected. (b) The set $A$ connot be expressed as the union of two nonempty sets that are separated in $X$. (c) There do not exist $U,V?T$ such that $U?A?=?,V?A?=?$, $U?V?A=?$, and $A?U?V$.